# ON SINGULARITIES OF FLAT AFFINE SYSTEMS WITH N STATES AND N-1 CONTROLS

Séance du mercredi 5 juin 2019, Salle L224, 14h00-15h30

Jérémy KAMINSKI, Holon Institute of Technology, Israël

We study the set of intrinsic singularities of flat affine systems with n−1 controls and

n states using the notion of Lie-Bäcklund atlas, previously introduced by the authors.

For this purpose, we prove two easily computable sufficient conditions to construct

flat outputs as a set of independent first integrals of distributions of vector fields,

the first one in a generic case, namely in a neighborhood of a point where the n − 1

control vector fields are independent, and the second one at a degenerate point where

p − 1 control vector fields are dependent of the n − p others, with p > 1. We show

that the set of intrinsic singularities includes the set of points where the system does

not satisfy the strong accessibility rank condition and is included in the set where the

distribution of vector fields, introduced in the generic case, is singular. We conclude

this analysis by three examples of apparent singularites of flat systems in generic and

non generic degenerate cases.

Jérémy KAMINSKI, Holon Institute of Technology, Israël

We study the set of intrinsic singularities of flat affine systems with n−1 controls and

n states using the notion of Lie-Bäcklund atlas, previously introduced by the authors.

For this purpose, we prove two easily computable sufficient conditions to construct

flat outputs as a set of independent first integrals of distributions of vector fields,

the first one in a generic case, namely in a neighborhood of a point where the n − 1

control vector fields are independent, and the second one at a degenerate point where

p − 1 control vector fields are dependent of the n − p others, with p > 1. We show

that the set of intrinsic singularities includes the set of points where the system does

not satisfy the strong accessibility rank condition and is included in the set where the

distribution of vector fields, introduced in the generic case, is singular. We conclude

this analysis by three examples of apparent singularites of flat systems in generic and

non generic degenerate cases.